It could be many things.
The first thought is a bad power supply for the one that doesn't turn on. But it could also be a bad switch on the case or switch connection is loose on the motherboard.
If you have another power supply, I would suggest trying to swap to a known working power supply to see if that is the issue. It is possible to short the switch connection on the motherboard to find out if the issue is with the case.
The one that starts and then shuts off could be a bad power supply as well. It all depends on how far it gets into the start up sequence. It could also be a bad motherboard. Bad capacitors on the motherboard can cause shut downs or not starting.
Since you have already opened the case to blow out dust, check to see of the capacitors are bad. You can tell by looking. Here is a pic of good and bad ones.
There was a period of time where there were a lot of bad capacitors and an Emachine is probably from that time period so I am guessing you Compaq is as well. It is possible to replace capacitors if you have the time and tools and nothing else to do. But for old machines, I would suggest recycling them and buying newer replacements. You can pull the hard drives and rescue any files on them.