It seems to me that the end mill is the weakest link at 1/8″ diameter…
Yes, sorry, I inadvertently complicated the discussion. The rigidity of the machine has nothing to do with your current problem.
I guess the three parameters I need to test are feed rate, depth of cut and spindle speed.
Exactly right -- generally referred to as "feeds and speeds" in the CNC community, though as @Ryan mentioned, number of flutes, and diameter of cutter are important too. (Really, in the end, the one most important measurement is chip load; your feeds and speeds, and bit geometry all affect chip load.) A "good" feed rate will also minimize bit deflection, which also shortens tool life. And finally, the bit material affects things. High speed steel will have different ideal speeds than carbide, which will be different from a coated bit. (If you're just getting started and only cutting wood, foam, plastic, wax, etc. just stick to carbide and don't worry about it.) All these nuances is why I start with a calculator to get me in the ballpark, then tweak it with multiple cuts to dial it in.
For tinkering, just googling settings that others have used for the material and cutter is fine - don't complicate it unless you need to.
If I understand you correctly, my settings were too conservative on feed rate/depth of cut and too aggressive on spindle speed.
That
was my take... but in my comment above, and as @Ryan pointed out, I missed that you were using a single-flute cutter. Based on a single flute cutter, your speeds and feeds seem more reasonable to me. Of course that leaves me scratching my head as to why you're getting a horrible screeching noise. I'm not very familiar with single-flute cutters, so if it were me, the first thing I'd try is a different cutter (2 flute), but that's mostly familiarity / laziness on my part.
The bearings in your router are good, right? (Most likely it’s the bit, but worth clarifying just in case, as the solutions are totally different. 
If your router was working fine before you dropped it in the MPCNC, I’m sure it’s fine…
Also, FWIW, in my experience 1/4" bits are much less likely to scream than 1/8".
Another thought: how far from the collet was the tip of the bit? In general, the shorter the exposed bit length, the more ridged the bit will be, and the less likely to deflect and/or chatter. If you can, choke up on the bit some more.
I’m new to the MPCNC, but have been using a hobby CNC machine (Shapeoko 3) for about 18 months, so I’ve learned a lot, but have a lot left to learn. (I don’t want to present myself as an expert - I’m certainly not.) However - here’s my take on rigidity: as long as you’re not going crazy fast, the biggest thing you will see with pushing the machine too hard is minor inaccuracies in cuts, especially in natural materials like wood. Cut a small (4-6") circle-diamond-square test, and put your calipers on it to see how close to specification the shapes end up. A well-tuned machine running safely within its rigidity limits will be accurate within a few thousandths. I suspect if you start pushing it too fast, you’ll see odd inconsistencies in the measurements… for example, the measurement of the diamond varying along it’s length as the middle assembly flexes slightly under the loads. If you don’t care about inaccuracies in the thousandths of an inch and/or don’t own a caliper, then don’t push the limits with your feed rate, and don’t worry about it. I don’t want to over-complicate a fun and rewarding hobby - you can make awesome stuff without ever knowing whether your machine is accurate to .003". And like what @Barry said, if you can see your rails flexing while cutting, you’re way behind the limits of the machine. Then you’re going to be measuring your inaccuracies in the .1" increments, not .001".
As @Barry pointed out, a much more likely negative outcome from pushing the machine too hard is the steppers losing steps. This is worth reading up on if you aren’t familiar with the term, but in short, the steppers miss movements because they don’t have enough torque to execute the instruction sent to them, and you get accumulated drift as they lose steps – completely ruining your part.
