I will open by saying this: There are no rules for Crouching in the core ruleset. There is Standing and Prone, with nothing in between. Thus, any rulings on crouching are entirely up to your DM.
So, the ultimate answer is this: Ask your DM. It's their call.
However, I will offer some advice based on what I've done in the past, simply rooted in what makes logical sense in allowing Crouching as a rule for D&D 5E.
What's the difference?
It makes logical sense to say that there is a difference between Crouching and Hiding, for the same reason that there's a difference between Dropping Prone and Hiding. One is a change in your character's stance, the other is an attempt to get opposing creatures to lose track of where you are.
To start with, we need to determine what you mean by crouching. We'll look at two possible examples.
- Dropping to one knee. This is a posture from which you could still meaningfully engage in combat...whether that's firing a bow, gesticulating for spells, or striking with a melee weapon.
- Sitting on your heels. This is what most people think of as 'Squatting' and is where you fold your legs up and put your butt down by your ankles. In this position, I imagine you would be significantly hindered from doing anything combat-like because your legs are now in front of your Torso.
For the sake of keeping things simple, I'm going to assume that races generally have a similar body length proportion to humans. This is not strictly true (see: Dwarves) but is a necessary concession to making any sense of this question.
In Case 1, by 'taking a knee' you are reducing your height by an amount equal to the length of your lower leg and foot. On an average person, this is roughly equivalent to 25% of your total height.
In Case 2, by squatting all the way down, you are reducing your height by an amount equal to the length of your entire leg. On an average person, this is roughly equivalent to about 50% of your total height. You could get to about this same height by taking a knee, then hunkering your torso down as low as you can get it.
So, by this we know that you have two options for getting behind a 3' wall. One of them leaves you fairly combat ready and reduces your height by 25%, the other should impair your ability to fight (Disadvantage, most likely), but reduces your height by 50% (And kneeling and hunkering would block you from doing much of anything unless you straightened up first).
In general, a person's shoulders are not going to be equivalent to 50% of their height...so assuming a 3' square block, the height of a character is the limiting factor.
To determine the heights of typical characters, there are tables in both the PHB and in Volo's Guide. Pages 121 and 120 respectively. These are NOT strictly binding, they are there to give you an idea of what is normal for these races. If you want to ask your DM if you can be a Goliath with dwarfism who is only 3' tall...go ahead and ask.
Assuming Minimum Height from those tables (you rolled a 1 on any rolled dice), we can draw the following conclusions.
Total Cover without Crouching
Halfling and Kobold
Total Cover on One Knee
Gnome (too tall to stand by 1 inch), Dwarf, and Goblin
Total Cover while Squatting
Everyone Else except a Bugbear, Firbolg, or Goliath.
Again, this is assuming that everyone from these races is as short as possible according to the tables. Simple rule is: If you're more than 6' tall, you don't fit behind the Minor Conjuration without going Prone.
What do you need to do
For this, I can only offer a guideline and recommendation based on what I would do.. You can drop Prone for Free, then stand up by using half your Movement. Based off that, it makes sense that dropping to a crouch would be equally free.
From there, my personal ruling would be that rising from Kneeling consumes 5' of movement, rising from a squat consumes one quarter of your Movement. But again, this is only my personal recommendation on how to handle it.
I use this in my games, and it has worked out fairly well. Had a Wizard who was short enough to squat behind the wall, and would just stand up to cast, then crouch down again. It only backfired when his 'cover' took a hit for him. It's an Action to use Minor Conjuration, and it disappears if the conjuration takes any damage...