Digging around, I can't find a definitive Monster Manual based answer here. But, there is a little piece from one of the published adventures for D&D 5E.
From Out of the Abyss (Spoilers, obviously)
Taking place in The Oozing Temple encounter, which is steadily flooding with water,
we get this little snippet:
Development: If Glabbagool is with the party, the intelligent gelatinous cube floats upward as the water rises and squeezes through a crack in the ceiling to escape the flooded temple and remain with the characters.
The only other possible answer I can give you is rooted in Science, which has a shaky relationship with a Fantasy world anyway.
In order to float, you must have a lower density than the stuff you are trying to float in. Average density is mass divided by volume. Of all things in D&D, a Gelatinous Cube is literally the easiest possible thing to compute the density of.
We know that an average gelatinous cube is 10 feet on a side...1,000 cubic feet. And, according to the 3.5E SRD, a Gelatinous Cube weighs 15,000 lbs. Divide that out, and that gives us a density of 15 pounds per cubic foot.
Convert to metric for comparison purposes and we get 240.277 kilograms per cubic meter.
Fresh Water has a density of 1,000 kilograms per cubic meter. (Salt Water is even more dense)
Thus, a Gelatinous Cube has a lower density than water. According to RL Science, it should float, and float pretty high in the water, too.
The move action and the Engulf's standard action are separate.
By virtue of the comment:
"The gelatinous cube merely has to move over the opponents, affecting as many as it can cover"
...it is apparently allowed to move into the spaces of opponents. If any creatures it wants to engulf is in the covered space at the end of it's move, it can then take a standard action to engulf whatever creatures are in the target area. (By changing the permeability of it's membrane, maybe?) If they succeed on the reflex save, they are instead pushed back or to the side (opponent's choice).
Creatures that fail are now "engulfed" and are thus in a legal positioning at the end of the cube's turn. Creatures that succeed on the save are "pushed back or to the side", and are also thus in a legal positioning at the end of the cube's turn.
Should it not choose to engulf, then it would be subject to the standard movement rules and have to end it's turn in a legal space.
Example:
Cube's turn.
Cube decides to take engulf action during it's turn as snacks are in range.
Cube takes move action. This move may end in squares overlapping opponents if and only if it is going to take an engulf action during this turn. This movement will trigger AoO's only if the movement would normally trigger by the AoO rules.
Cube takes standard action to activate engulf. This triggers the option for Reflex save OR AoO per the Engulf ability, which may have nothing to do with movement triggered AoOs.
Everyone ends up either inside the cube taking damage, or pushed back or aside to a legal position and not engulfed.
End of Cube's turn.
Best Answer
The Forgotten Realms wiki page on gelatinous cubes has the information you are looking for in the "Ecology" section: