You mentioned in your answer that you wanted to keep electric strings on your acoustic electric guitar. You did not provide a reason for this.
Electric Guitars usually have magnetic pickups which work by picking up on the magnetic strings vibrations. There is a pole piece under each individual string.
In order for these magnetic pickups to react to and process the vibration of the guitar string, the string must be a type metal that responds to a magnet. Electric guitar strings have a steel core and the wound strings are wound in nickel instead of bronze. Nickel wound strings do not produce the resonance preferred in an acoustic guitar and tend to sound dead when played acoustically. This is not an issue with an electric guitar because it is intended to be played using an amplifier to produce the sound the listener hears.
In contrast, most pickups for acoustic electric guitars are of a non magnetic variety and detect vibrations either in the saddle (under saddle piezoelectric)
or the top (body sensing pickups also using piezoelectric technology). Some systems designed to allow an acoustic guitar to be amplified use a microphone inside the body to literally mic the acoustic sound.
Therefore you can use acoustic guitar strings on acoustic guitars with non magnetic pickups. Acoustic guitar strings have bronze windings. The bronze or phosphor bronze winding material does not respond to the magnets in magnetic pickups.
Some acoustic electric guitars come with factory installed magnetic pickups and there are a number of magnetic sound-hole pickups that can be mounted on most acoustic guitars to allow them to be amplified.
If your guitar uses one of the less common magnetic pickups, then you will get better results from the amplified sound using electric guitar strings. However, when you play the guitar acoustically, it might not sound as good.
The high e-string that comes in an electric set and in an acoustic set of strings is made of the same material - usually steel. The only difference is in the string gauge which is often lighter in electric sets to allow for more bending. But you will find electric sets with the same string diameter as some acoustic sets.
The gauge of the steel string high e used to replace the broken string should be consistent with what would be used in combination with the gauge of the remaining strings. This will maintain a consistent feel and playablity across all 6 strings.
Unless the other 5 strings are almost new, it would probably make sense to buy an entire set. The type strings you choose is highly personal and the only reason to limit your selection to electric strings is if your guitar does in fact have a magnetic pickup. Otherwise you may get better sound overall (both acoustic and amplified) by choosing an acoustic set which comes in a multitude of varieties.
The slots on your nut (at the base of the headstock) will limit the diameter of strings you choose for your guitar. Too thin and they will move around too much inside the slot. Too thick and they will not fit in the slot. Most nuts have slots that will accommodate a range of the more common string gauges.
You should probably visit a guitar shop and discuss the pros and cons of various type strings, play some guitars in the shop with different string types and if you can afford to - have a qualified guitar tech do a full set up on your guitar based on whichever type strings you decide to go with.
Good luck and keep it fun.