A little while ago I had a couple of GU-10 bulbs burn out. I went and replaced and a couple weeks later they blew out and tripped the circuit to the whole bedroom. Once the circuit was flipped back All the lights worked but the 3 recessed GU-10 lights. I replaced the bulbs and they still don’t work. What else could be the issue?
Hmmm...sounds like a bulb itself shorted (true short) and tripped the breaker. It's rare for bulbs to do that, but it happens. Then a connection someplace failed and now you have an open circuit. Are there other lights on the same switch that control the GU-10s? If they work, then it's not the switch. But if just the GU10s are on a separate switch here's what you do: With the power off start with the switch, pull it out and check it for damage. If a "back stab" switch, re-wire it using the screws. If you can't get the wire out of the back-stab, it's OK to cut it off flush with the back of the switch, re-strip the wire and use the screws on the side to re-wire. The switch itself may have gotten fried. After you have removed the switch, if you have a multi-meter, set it on ohms and test resistance across the screws. You should get almost infinite resistance when "off" and very low or zero resistance when "on". If you don't have a multi meter, you could simply replace the switch with a high quality unit, they aren't expensive.
Never use the "back stabs" on switches or outlets. They are a well known point of failure.
Anyway, reconnect the existing or new switch and try it. If still not working, you'll just have to get into the fixture boxes and check connections.
Again, the fact you had a true short that flipped a breaker means a poor connection someplace got fried.