I'm DMing a game and one thing that arose was combat in a 5' wide corridor. The bad guy dropped the lead pc to 0 and I moved him into that downed players space to attack the next in line. My players all put up a fuss that you can't occupy the same space. My counter point was that is normally true but that pc is no longer a threat or can take actions. By the rules as written which seem vague can someone occupy the space of an unconscious person? My whole ploy was to engage and if a pc drops occupy that space and keep pushing down the hallway.

Well ultimately it is up to the DM, but remember any house rules you employ are usable by both NPCs and PCs.

According to the rules (https://www.dndbeyond.com/sources/basic-rules/combat#MovementandPosition) creatures can move through another creature's space, but cant stop there (like to make an attack). Being prone or unconscious doesn't seem to change that (being dead would).

It is really easy to imagine stepping over a fallen enemy/ally.

I would say yes, you can step into the same space as a prone creature, but it counts as difficult terrain. If that creature is conscious and friendly, I would require some form of communication, or risk their movement interfering in the standing creatures actions; if conscious and enemy, I would give the prone creature advantage in knocking the standing creature prone.

From experience on both sides, yes, you can step over/on a prone person. I would consider it difficult ground, especially in a combat situation, where you have more to watch than just your feet.

Stepping into the space of an awake, unfriendly, and resisting creature, even if they were prone, would not be so easy. I five foot square is a pretty big space compared to a human, standing or prone. But if everyone is moving around and jockeying for position it is a reasonable space to consider covered and controlled. The effect of everybody moving around is an easy one to overlook when looking at a map with static grid squares and counters or when everybody is friends standing in a hallway to see how it looks.

My whole thing is I wanted him to end his turn in the square and have him stand his ground. I'm thinking maybe about disadvantage on any attacks he makes to reflect trying to move around with an unconscious body underneath. But I see no reason why a character pc or bad guy can't occupy the same space if they are down and out.

It would be simpler to just let it happen if the creature is dead/unconscious. Even if it seems "realistic" it's not necessary to create new rules for it. It also plays as heroic if a PC stands over his friend defending them by standing over their body - I wouldn't want to give them disadvantage for that.

Another option to consider is dragging the character out of the space first.

Might be a little anti-climactic to KO a PC then drag them back 5 feet to step over them. You could narrate it as they step over and kick(shove) the creature back a space.

It will use 1 attack of a multiattack to do so (not actually an attack, no failed death saves).

I like the shove idea. and if the PCs give you grief, stab the dying bastard, so that he's no longer a creature, and carry on your merry way.  Lesson learned.

RAW no, logically yes. The problem comes when he is healed and gets up. Does that force the enemy out of his space?  Is he required to crawl away to another square in order to stand?  If other squares are blocked is he just stuck on the ground below his enemy? 

When you change the rules because it makes sense, that can sometimes require you to create more houserules to deal with issues it creates.  The easiest solution is to just kick the unconscious person out of the way as suggested above.

If you knock out enemy creatures rather than kill them, you can make a small fort with their unconscious bodies.