First, there shouldn't be a problem using a brown chicken proper (and you shouldn't have to present evidence that there were brown chickens in real life in period) because there are brown chickens in period heraldry. Here're a few examples, just in case you need them.
The birds I'm seeing as supporters all seem to be standing on one foot and gripping the shield with the other (sometimes with the gripping foot clearly visible, sometimes with it largely obscured by the shield), like these:
I'd recommend you go with that posture for your chicken.
Here are some more chickens from heraldic contexts. They should (along with the brown chickens above) give you an idea of the range of chicken types seen in period heraldry, and maybe one of them will work as a starting point for you.
Some of these are from achievements associated with names that sound like words for "hen" or "rooster" in the languages of the books in which they appear. These are cants (visual puns, which were quite popular in period heraldry). I don't notice anything distinguishing the birds canted with "hen" names from those canted with "rooster" names. Therefore, I think it likely that in period heraldry a chicken was a chicken was a chicken, and that you don't have to worry about trying to make the sex of yours obvious.
If you are interested in seeing an undisputed hen, I found one from a 12th-century manuscript. But it's not from a heraldic context, and it pre-dates the use of supporters in heraldry by some centuries.
One last note: Wing position was one of the things period heralds used to distinguish one type of bird from another, and the overwhelming majority of chickens in period armory have their wings folded. But there are some shown with their wings either spread out to their sides or stretched up behind their heads, like this:
Since we know it was done with chickens, and we know it was done with supporters (see the first and last examples above), it's reasonable to argue it might've been done on a chicken that is a supporter. Just be aware, if you decide on spread wings, that it is uncharacteristic, so you'll have to be extra careful to make it very obviously a chicken in every other respect.
Bulls used as supporters seem to all be standing on both hind feet, with the forefeet touching the edge of the shield, posed like the hands of a gameshow model (near low, far high), and the tail curved down along or between the rear legs.
I think any of these could reasonably be turned into a cow with the addition of an udder. The horns are pretty small, and they do a lot to clarify the bovine character of the critter, so I'd recommend you keep them.