You can swim any dang stroke you want to, free style, doggy paddle, side stroke, breast stroke, back, heck if you have enough energy do the fly. As for continuous, well, if I remember correctly, you had to continuously keep moving and not touch the bottom, so I would imagine that you would have to continuously keep using one of the above mentioned strokes. If you are talking in regard to "pace", yeah, you are not in the olympics, you are just swimming to get under the required time limit (been too long since I did it, 45 min or something, wasn't it?), which is a ridiculous amount of time to swim that in. So whatever stroke you are using is going to be at a slow pace.
Why not go practice?, it amazes me the number of people (that go into the Navy no less) that are uncomfortable in the water. Then you throw a flight suit on them and they have to swim, next thing you know, they are on their 5th attempt at the mile swim. Not saying all weak swimmers will end up with that, but the day my API class did the mile swim, there were 6 or 7 rollbacks from a previous class, of which 3 failed, again.. as for one person failing, there probably is that one person out there somewhere, but the Navy and the program bend over backwards to train and help the weak swimmers along. But if you are that weak swimmer, and you know if you are, do you want to spend that extra time on remedial swim there in API, or improve on a weakness now in your spare time? I thought the motto was drop the donut and PT that ass or something to that affect?
A mile swim won't give you a rash that will kill you (not like it is Dive school where you get sugar cookied and have nasty rashes from the salt and sand after the first week), but some guys just grab the smaller flight suits and walk away from the mile swim with nice reminders. You don't need to cake it on, but it doesn't hurt. Think smarter, not harder...