Since the only fuel available on an airliner is kerosene, the APU is a kerosene-powered jet. Using a different and maybe more efficient type of engine would require an independent fuel system, which does not make sense for the small savings possible.
From the 737.org page on APUs:
There are many different APUs available for the 737. The most common
is the Garrett GTCP (Gas Turbine Compressor [air] Power unit
[electrics]) 85-129. This was standard for the series 1/200 but when
the -300 was introduced it was found that two to three times the
energy was needed to start the larger CFM56 engines. Garrett produced
the 85-129[E] which had a stretched compressor, ie the impellers were
lengthened and the tip diameters increased. When the 737-400 was
introduced, even more output was required and Garrett produced the
85-129[H]. This has an Electronic Temperature Control which limits hot
section temperatures depending upon demand and ambient temperatures.
By 1989 the 85-129[H] was the most standard APU in all 737 models,
although there are actually 14 different models of the 85-129 in
service with 737s.
To answer your question about the 737 NG, it continues:
Other APUs available are the Garrett GTCP 36-280(B) and the
Sundstrand APS 2000 on the 3/4/500; and the Allied Signal GTCP 131-9B
for the NGs. The main difference between them is that the Garrett is
hydro-mechanical whereas Sundstrand and Allied Signal are FADEC
Be sure to load the 737.org page, it has lots of pictures and data on the APUs.