How to turn pulsating DC (square wave) from a 555 IC timer into an ac square wave

You want a very basic AM transmitter. The old Knight-Kit C100 walkie-talkie has a very simple design. It used a super-regenerative receiver, to minimize parts count. It had one crystal, to set the transmitting frequency, and three transistors. You could probably simplify that, to make just a transmitter. The schematic USED to be available on the Web, but it appears to have vanished.

OLD copies of the Radio Amateur's Handbook should have circuits that could be adapted to a simple transistor AM transmitter.

If I were going to try to throw something together, I'd probably go DSB rather than AM, and use an SA612AN (NE602 replacemement). Instead of a VFO, I'd hang the crystal on the SA612AN, and I'd probably try a plastic 2N2222A, either barefoot or driving a 2N3866. I'm not sure what I'd use for an audio preamp. Here's a more elaborate transceiver, that illustrates the basic principles.

Now for the bad news. There's no way you're going to power this from a CR2032 coin battery. Look at camera batteries, like a Kodak 2CR5 6V lithium battery.

You'll need to couple the square wave with a capacitor to the transformer because standing DC voltages will just cause heat but there is no problem feeding a transformer with a square wave in principle.

In practice, if you fed a 60Hz "power" square wave to a transformer, the higher order harmonics in the square wave would mean that a regular AC power transformer wouldn't be as efficient as being fed with a sine wave.

A 24V 6 A supply is capable of providing a power of 144 watts and your output requirement is also 144 watts so you need to be aware that you might only get 90% efficiency and expecting 144 watts from the transformer output is a little naive.