The only safe method is to bring all 4 disks to commercial HDD restoration center, they can raw-read disks and construct a volume with full copy of data on spare HDD(s) using special programs. BTW, configurations of an even number of disks in RAID-5 is not recommended at all, including reason of what happened with you (danger of 'array split').
I can hardly believe that there is no simple way to change a label or something to get the system to recognize it's own drives. Nothing has changed on the drives that I know of. I don't know how RST recognizes it's own drives, but this so far is the worst situation anyone's RAID software has put me in. I feel so screwed by using the Intel software that came with my motherboard. If all it takes is a BIOS change back and forth from raid to destroy a system, how did Intel let this into production? And BTW, does no one at Intel follow this site? I expected that someone from the group that developed RST would have seen this and looked into this problem by now.
It is quite odd that you state that an even number of disks can be problematic, as the commercial RAID systems that I have supported had 30, 60, and 80 drives, respectively, and non of them ever had a problem along these lines.
That was RAID5 arrays with such numbers? More likely RAID-10 or 50. Even numbers in plain RAID-5 not recommended mostly for performance reason, cause we have a parity part and I/O page sizes are power of 2, so there is a risk that end of IO will require writing two stripes instead of one.
As JFFulcrum stated if you want to make sure you save all your HDD's information you will need to take your disks to commercial HDD restoration center.
On the other hand, you are able to restore the RAID 5 with four HDD's by changing the BIOS setting to SATA> RAID Mode. and reinstalling the Operating system in A Raid ready system.(All information will be lost on the HDD's) In order to complete this task you could try the "Ctrl"+"I" key combination at the POST screen in order o set the RAID configuration again.
You may find these articles useful:Intel® Rapid Storage Technology (Intel® RST) — System Will Not Boot to RAID Volume and Intel® Rapid Storage Technology (Intel® RST) — Troubleshooting RAID Volume Issues