If you have a dremel tool use the cut off wheel and cut a flat blade screw slot in the  screw or what ever you need and use the screw driver to remove it..

My favorite tool is very similar to this one. For only $7 it's worth buying, especially if you're buying used engines that someone else may have damaged before you get them.
You chock up the engine so it's held tight, use the bottom side of the tool pictured. Slow speed with a good amount of pressure, spins them right out every time. Obviously the screw is trash after that, but no harm to the engine.

https://www.amazon.com/gp/aw/d/B01NGTLM40/ref=mp_s_a_1_5?ie=UTF8&qid=1516981320&sr=8-5&pi=AC_SX236_SY340_FMwebp_QL65&keywords=damaged+screw+extractor+set&dpPl=1&dpID=51REgeV2qPL&ref=plSrch

can we see a picture of what we are working with

If you have a dremel tool use the cut off wheel and cut a flat blade screw slot in the  screw or what ever you need and use the screw driver to remove it..

Or just cut off the bolt heads & use new bolts.

It's kinda tough to mount an engine with the muffler on it.
Depending on the size engine, these should be 2.5MM or 3MM bolts.

In future should this arise, it is helpful to grab your phone or similar and take a photo of what you are dealing with and post it here to let others know what you are dealing with. It would help to know what OS engine you are talking about and if the muffler is still attached. Some mufflers attach with the screws entering the engine from the muffler side (they go through the muffler and into the engine. Others go through the engine and then into the muffler.

Obviously if it is the prior then one screw will be next to impossible to get at with a drill or any other implement as the muffler gets in the way. If it is the latter then a different techniques are called for. For both the path of least resistance (If the muffler is still attached) is to get rid of the screw heads. If it is the type where the screws pass through the muffler lugs and screw into the engine, Taking a file or other method to remove the screw heads, allows you to remove the muffler and you are left with about a quarter inch of screw sticking out of the engine to grab with vice grips and carefully wiggle the free.

If it is the type that pass though the engine first then screw into the muffler, knocking the head off lets you remove the muffler and you have the remainder if the screws sticking out of the muffler. As these screws are relatively small either using a drill or a file to remove the heads is the path of least resistance. Actually this is usually the best approach for getting small screws out of just about anything.

Kind of leaning toward cutting off the bolt heads as I can then pull off the muffler, remove the studs and find new bolts.

Photos here of the problem.

looks like drilling them off'd be easy . 3 mm or 1/8 drill . And a clout hammer . 

I've used needle nosed vice grips successfully on the exact same situation you're in. Warm the engine a bit with a heat gun. Some spray PB Blaster won't hurt.
If you can file a flat on the screw head first, that won't hurt either.
Plan B is screw head removal. Phillips type screws tend to center a drill bit if you're careful.
Best of luck. You're a good builder, you'll be able to do this.

Use a round rotary file to clean out the damage in the screw slot or you'll break the drill bit. Then drill with a left hand bit, nice and slow and many times they'll walk right out as you're drilling them.

Here's a 3mm LH bit: https://www.ebay.com/i/273018637564?chn=ps

Well, fellas.  I just ordered a set of the JIS screwdrivers.  More to follow.  Thanks.

JIS did not work.  Looks like an ez out?

Oh, Ty replied at the same time. I'd take him up on that, mostly just to go hang out with him!