Linux Software RAID1 Rebuild Completes, but after reboot, its degraded again

You need to install GRUB to the MBR of both drives, and you need to do it in a way that GRUB considers each disk to be the first disk in the system.

GRUB uses its own enumeration for disks, which is abstracted from what the Linux kernel presents. You can change which device it thinks is the first disk (hd0), by using a "device" line in the grub shell, like so:

device (hd0) /dev/sdb

This tells grub that, for all subsequent commands, treat /dev/sdb as the disk hd0. From here you can complete the installation manually:

device (hd0) /dev/sdb
root (hd0,0)
setup (hd0)

This sets up GRUB on the first partition of the disk it considers to be hd0, which you've just set as /dev/sdb.

I do the same for both /dev/sda and /dev/sdb, just to be sure.

Edited to add: I always found the Gentoo Wiki handy, until I did this often enough to commit it to memory.

First off, check the physical hardware like connections, cables, and correctly-seated cards. For the disk itself, check SMART data on /dev/sdb to make sure the disk itself is not failing out periodically. Western Digital Raptors are fast but prone to failure, I've had one fail on me out of nowhere (not even SMART data predicted it). Use smartctl to read the SMART data and run tests. It comes in the smartmontools package:

apt-get install smartmontools

Pull the data and look for anomalies or errors logged:

smartctl -a /dev/sdb

Finally, run a manual self-test, which will take about 2 minutes. long can be substituted for short, and is more thorough, but it takes far longer ("tens of minutes"):

smartctl -t short /dev/sdb

Once the test is done, review the results:

smartctl -l selftest /dev/sdb

If it all comes back clean, you can move on to debugging the mdadm stack.

Your partition arrangement is a bit strange on the RAID devices. If those devices will be dedicated to RAID, you don't need a partition table at all. Assuming there's no data, you'd be well advised to keep it simple and use the block devices directly. In fdisk they would show up like this:

Disk /dev/sdb: 300.0 GB, 300069052416 bytes
255 heads, 63 sectors/track, 36481 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Disk identifier: 0x00000000
Disk /dev/sdb doesn't contain a valid partition table

To eliminate any issues with partitioning and start from scratch, just dd some zeros at the beginning of each disk;

dd if=/dev/zero of=/dev/sdb count=128 bs=4096k

Repeat for /dev/sdc. Create the array using those two devices:

mdadm --create --raid-devices=2 --level=raid1 --bitmap=internal --assume-clean --name=RAID1 /dev/sdb /dev/sdc

Don't forget to check dmesg for any disk-related output!