As above

Hi Tally! Welcome to the forums!
To your question, yes, of course you can eat oats. But maybe you won’t WANT to if you consider the situation more fully.
If you are doing this diet to control your blood sugar (diabetic or non-diabetic) and to lose weight, choosing oats as a meal might not be your best choice.
Gluten free makes no difference (and is probably WORSE because gluten is the PROTEIN in oats) in how your body will deal with the carbohydrates in the oats. Once your body breaks down the oats, it ends up as a simple sugar and acts to raise your insulin hormone response.
Oats are mostly carbohydrate, and with the gluten removed they are pretty much ALL carbohydrate. They do have some fibre but it doesn’t slow down the digestion enough to really be “slow release” (I am assuming here that you are eating rolled oats).
If you were eating the WHOLE oat, unprocessed, uncut and not rolled, you might be ok but the cut and rolled oats we usually eat are quickly digested and quickly raise your blood sugar levels.
A very important point is that to lose weight and control our appetite, we absolutely MUST get our insulin levels down.
Your insulin is a hormone that works to store fat and the way it does that, is it pulls blood sugar out of your bloodstream and packs it away in your fat cells. It fills up your fat cells AND then it leaves you with low blood sugar. Your body “reads” that low blood sugar as trouble — serious trouble — and you then want, or need, to eat to resolve that low blood sugar.
As long as your insulin hormone is high, you cannot ACCESS the many calories you have stored in your fat cells. Your insulin has acted like a lock on the fat cell doors and no fat can flow out into your bloodstream.
Now you are feeling starving and actually, on a cellular level, you ARE starving. Your body isn’t misleading you, your body is absolutely low on fuel. It’s no joke, you’re hungry, weak and feel clammy and weird — classic low blood sugar symptoms.
That’s what oats will do for you.
As an alternative, I recommend eggs (nature’s perfect food according to a famous neurologist) and meat and cheese for breakfast. Add in some cream in your coffee, some sautéed mushrooms and peppers and you’ve got yourself a decent meal!
I know it’s hard to accept. It took me a long time to realise that all the foods I had been told were healthy were not healthy for me. But TRY it!!! Just give it two weeks going very low carbohydrate (try to go as low as 20 grams per day, or at least below 50) and see how amazing you start to feel — no more low blood sugar, no more crazy appetite, brain working at full speed — it’s truly a revelation.
This diet changed my relationship to food forever and it can do that for you too.
Maybe ditch the oats?

Just to point out that oats are naturally gluten free. They put that on the packet so they can charge more but it also guarantees that it has not come into contact with gluten during the processing in case you are really intolerant.