Can I take my ReadyNAS drive in Raid1 and plug it straight into new different machine?

Yes and no.. If you use a hardware RAID, you can only use the drive in the same model (or chipset) RAID controller. If you use software RAID, then you can swap drives happily (but not, obviously, swap Windows to Linux software raid).

As you have it running in a RAID 1 configuration, go right ahead and swap and see. Once you take the drive out, you'll replace it and the NAS will rebuild the array. If it doesn't work in the new enclosure, replace it in the NAS and it'll rebuild the array.

This is one of the major drawbacks to some of the NAS RAID devices for home. An example would be my situation. I own a Thecus 2100 NAS device running 2x320GB drives in RAID1. The Thecus device itself failed along with one of the 2 drives. My only solution was to purchase the exact same model to recover the data. There was no other way for it to be done. Even tried 2 other Thecus RAID models. So the answer is going to typically be: it depends on your original setup. At the very least, you'll likely need a similar RAID support to plug into so just hooking it up via a USB enclosure is not a likely option.

Others may have more experience and may be able to provide a more solid answer about what setups would allow that type of recovery. I would also be interested in this option for recovery of downed RAID1 NAS setups.

RAID1 is computationally almost free, so software RAID1 is usually preferred in smaller setups. Linux (mdadm) and Mac (Disk Utility) systems can use half of a pulled-apart software RAID1 disk without issue; it is recognized as such. You might not be able to write to the disk without deliberately remounting the disk read/write, but at the very least you can read from it to salvage the data thereon.

Windows sucks, and so does its software RAID. ("Dynamic disks"? No thanks.) I'm not going there.

Updated re hardware RAID1: If you use the same hardware RAID adapter, then yes you can move the hard drives into another machine. If you don't, then you generally can't.