Instructions on how to modify the Minecraft.app for OS X to work with Java 7 (without the need for 6). See the instructions in the "ReadMe" section.

ReadMe - Instructions.txt

These instructions are for Mac OS X 10.7.3 and later (those that can run Oracle's Java 7).

This assumes you have the Java 7 JDK installed from: http://www.oracle.com/technetwork/java/javase/downloads/index.html

When you download and install Minecraft.app into /Applications, you can control-click or right-click on the Minecraft.app file and "Show Package Contents".

Inside the "Contents" folder inside, there is a "Info.plist" file, which is plain text. Open this "Info.plist" file in a text editor of your choice and replace the contents with the "Info.plist" listed below.

Then open the Terminal and run the following commands:
touch /Applications/Minecraft.app/Contents/MacOS/LaunchGame.txt

This will create a blank text file called "LaunchGame" inside the Minecraft.app/Contents/MacOS folder.

Open the file and paste in the contents of the "LaunchGame" listed below.

Then run these commands in the Terminal:
mv /Applications/Minecraft.app/Contents/MacOS/LaunchGame.txt /Applications/Minecraft.app/Contents/MacOS/LaunchGame
chmod +x /Applications/Minecraft.app/Contents/MacOS/LaunchGame

As a last step - you should be able to double-click Minecraft.app and now it will launch using Java 7 (assuming it's installed). If you get an error message about an unknown developer, you may either have to adjust your Gatekeeper security settings in OS X or (easier) if you're an admin, you can just control-click / right-click on Minecraft.app and select "Open" and it will let you open the file successfully. Once you've done this once, double-click should work from that point on.

Info.plist

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist SYSTEM "file://localhost/System/Library/DTDs/PropertyList.dtd">
<plist version="1.0">
  <dict>
    <key>CFBundleDevelopmentRegion</key> <string>en</string>
    <key>CFBundleName</key> <string>MinecraftLauncher</string>
    <key>CFBundleVersion</key> <string>1.0.1</string>
    <key>CFBundleShortVersionString</key> <string>MinecraftLauncher 1.0.1</string>
    <key>CFBundleExecutable</key> <string>LaunchGame</string>
    <key>CFBundlePackageType</key> <string>APPL</string>
    <key>CFBundleSignature</key> <string>????</string>
    <key>CFBundleGetInfoString</key> <string>MinecraftLauncher 1.0.1 © Mojang Specifications, Inc, 2013</string>
    <key>CFBundleIconFile</key> <string>favicon.icns</string>
    <key>CFBundleAllowMixedLocalizations</key> <true/>
    <key>CFBundleInfoDictionaryVersion</key> <string>6.0</string>
    <key>CFBundleIdentifier</key> <string>com.Mojang Specifications.Minecraft.Minecraft</string>
    <key>LSHasLocalizedDisplayName</key> <true/>
    <key>CFBundleDisplayName</key> <string>Minecraft</string>
    <key>NSHighResolutionCapable</key> <true/>
  </dict>
</plist>

LaunchGame

#!/bin/bash
SCRIPTPATH=$(cd $(dirname $0);pwd -P)
/usr/bin/java -Xdock:icon="$SCRIPTPATH/../Resources/favicon.icns" -Xmx128m -jar "$SCRIPTPATH/../Resources/Java/Bootstrap.jar"

Zulith, did you have Java 1.8.0 u40 and then upgrade to Java 1.8.0 u60? Perhaps /usr/bin/java is linked to u40 but it's already been removed and replaced by u60. It's trying to execute /Library/Java/JavaVirtualMachines/jdk1.8.0_40.jdk/Contents/Home/jre/bin/java even though it seems to use u60 in other areas.

When I run it in the terminal it just says
Error: Unable to access jarfile /Applications/Minecraft.app/Contents/MacOS/../Resources/Java/Bootstrap.jar
logout

[Process completed]
and when I open it in applications, it says
Process: launchd [430]
Path: /Applications/Minecraft.app/Contents/MacOS/launcher
Identifier: com.mojang.minecraftlauncher
Version: ??? (???)
Code Type: X86-64 (Native)
Parent Process: launchd [125]

Date/Time: 2016-07-18 21:02:19.275 -0400
OS Version: Mac OS X 10.7.5 (11G63)
Report Version: 9

Interval Since Last Report: 19350 sec
Crashes Since Last Report: 17
Per-App Crashes Since Last Report: 5
Anonymous UUID: F0E202FB-C38A-4FF6-8C62-2100CAC60FA9

Crashed Thread: Unknown

Exception Type: EXC_BAD_ACCESS (SIGSEGV)
Exception Codes: KERN_PROTECTION_FAILURE at 0x00007fff5fc01028

Backtrace not available

Unknown thread crashed with X86 Thread State (64-bit):
rax: 0x0000000000000055 rbx: 0x0000000000000000 rcx: 0x0000000000000000 rdx: 0x0000000000000000
rdi: 0x0000000000000000 rsi: 0x0000000000000000 rbp: 0x0000000000000000 rsp: 0x0000000000000000
r8: 0x0000000000000000 r9: 0x0000000000000000 r10: 0x0000000000000000 r11: 0x0000000000000000
r12: 0x0000000000000000 r13: 0x0000000000000000 r14: 0x0000000000000000 r15: 0x0000000000000000
rip: 0x00007fff5fc01028 rfl: 0x0000000000010203 cr2: 0x00007fff5fc01028
Logical CPU: 0

Binary images description not available

External Modification Summary:
Calls made by other processes targeting this process:
task_for_pid: 1
thread_create: 0
thread_set_state: 0
Calls made by this process:
task_for_pid: 0
thread_create: 0
thread_set_state: 0
Calls made by all processes on this machine:
task_for_pid: 673
thread_create: 0
thread_set_state: 0

Model: iMac5,1, BootROM IM51.0090.B09, 2 processors, Intel Core 2 Duo, 2 GHz, 2 GB, SMC 1.8f2
Graphics: ATI Radeon X1600, ATY,RadeonX1600, PCIe, 128 MB
Memory Module: BANK 0/DIMM0, 1 GB, DDR2 SDRAM, 667 MHz, 0x7F7F7F7F7F510000, 0x36345431323830323145444C335342322020
Memory Module: BANK 1/DIMM1, 1 GB, DDR2 SDRAM, 667 MHz, 0x7F7F7F0B00000000, 0x4E5431475436345538484230424E2D334320
AirPort: spairport_wireless_card_type_airport_extreme (0x14E4, 0x87), Broadcom BCM43xx 1.0 (5.10.131.36.15)
Bluetooth: Version 4.0.8f17, 2 service, 11 devices, 1 incoming serial ports
Network Service: Wi-Fi, AirPort, en1
Serial ATA Device: WDC WD1600AAJS-60M0A0, 160.04 GB
Parallel ATA Device: PIONEER DVD-RW DVR-K06
USB Device: Built-in iSight, apple_vendor_id, 0x8501, 0xfd400000 / 2
USB Device: Hub in Apple Extended USB Keyboard, apple_vendor_id, 0x1003, 0x5d100000 / 2
USB Device: Apple Optical USB Mouse, apple_vendor_id, 0x0304, 0x5d110000 / 4
USB Device: Apple Extended USB Keyboard, apple_vendor_id, 0x020b, 0x5d130000 / 3
USB Device: Bluetooth USB Host Controller, apple_vendor_id, 0x8206, 0x7d100000 / 2
USB Device: IR Receiver, apple_vendor_id, 0x8240, 0x7d200000 / 3

I'm having similar problems guys
Here's what my console says when I click on minecraft:
Last login: Wed Nov 16 10:45:02 on ttys000
FerdTrekampsMBP:~ ferdtrenkamp$ /Applications/Minecraft.app/Contents/MacOS/launcher ; exit;
2016-11-16 10:45:04.632 launcher[3581:146926] Claimed it found a path: /Applications/Minecraft.app/Contents/MacOS/launcher () 4096

Okay, finally got Minecraft working on my ancient Mac running OSX 10.7.5 (Lion, I believe, no I refuse to upgrade!).
The solution is to forget all of the above github, which did not work at all (missing bootstrap.jar, launcher exit, etc.), oh, except, do install the latest Java from Oracle (not Apple): http://www.oracle.com/technetwork/java/javase/downloads/index.html
What worked for me was to go to mc-launcher.com, click on Minecraft, click the Uncompressed MacOS jar file download. Click Skip Ad in the top right (the popups and main page are all spam!) and you will get TLauncher-MC.jar in Downloads. Double click it, open it, DO NOT install the java runtime for OSX from Apple if prompted, (install the one from Oracle), click Install and Enter, create an account by clicking the green plus sign (or add your existing mojang account), (I unchecked the elyby skins). Click Save Account. Click the Home icon in the bottom right. Then Install and Enter. That's it!

I'm using an early 2007 MacBook I've tried using this method on other apps and it bounces the app up once and then quits unexpectedly I think I know why correct me if im wrong but its a 64 bit app im trying to run and its a 86x64 system