[COLOR="Blue"]Is it one of those freezers that stays running?[/COLOR]
No

[COLOR="Blue"]If not, what could be the problem?[/COLOR]
You have to do more troubleshooting.

First what is the temperature in the freezer?
It should be between 0 and 5 degrees F.

If a lot lower it could be that the thermostat contacts are always closed so it keeps running.

If higher then try setting it to a lower temperature to see if it gets colder.

If it does not then remove the cover in the freezer so you can inspect the evaporator coils.

If they are heavily iced/frosted over then it is a defrost problem.
With the coils iced over the fan cannot draw air through them to cool it properly so the compressor keeps running trying to get to the set point temperature.
This should be fixable at a reasonable cost.

If there is frost in only one area (usually where the freon enters the coils) or it is patchy then odds are that it is a sealed system problem.
In this case it is probably not worth fixing due to the cost.

Be sure that the compressor has run for at least an hour before checking the coils. If you do this during or shortly after a defrost cycle you may end up with the wrong diagnosis.

I am not sure what is going on here.

If the control thermostat has an off position does the unit turn off?

If you move the thermostat to a warmer setting does the unit shut off?

[COLOR="Blue"]The evaporator motor makes a squeaking sound once in a while, could that be bad?[/COLOR]
Yes it could.
If the fan does not pull enough air through the evaporator coils it cannot cool the air properly so the unit/compressor has trouble getting the temperature down to set point.

Since your frost pattern looks OK I would also check a couple other things.

Check/clean the condenser coils under the unit.

Check the condenser fan. Also clean the fan blades if they need it.

Check/clean the kick plate vents and the rear cover vents.

Make sure that any interior lights shut off when the door is closed.

Check the seals on the doors.
Get a 2 inch wide strip of paper. place it between the door seal and the frame.
Close the door and then pull the paper out to check the seal.
Work your way around the door every 6 inches or so.
Check the hinge side of the seal to make sure it is not grabbing and twisting when you close the door. If it is then a light coating of white lithium grease or vaseline should solve this.

Unplug the unit and one wire from the thermostat.
Be sure it cannot short to anything else.
Then plug he unit in and see what happens.

Looking over our previous posts perhaps I have a communication problem.
It is not only the fan but the compressor stays on, is that correct?

Also by the sound of it this unit has done this since you got it, is that correct?

The reason I ask is that the only thing I can think of that would keep the fans and the compressor on with the thermostat disconnected is that someone has messed with the wiring.

I have looked around but unfortunately cannot find a wiring diagram for this unit so that you could trace the circuits and find where the problem is.

On older units the wiring diagram was sometimes glued to the back of the unit.
You really need this to go any further.