It should be noted that the original version of the game has a single achievement called "BETA Complete Me". In the Director's Cut this was expanded into a total of 3 achievements, each corresponding to 3 hidden puzzle rooms.

This is what you need to find. See that silky glossy border? Yeah, 3 of those fake walls with that. Let's get to it.

Complete the hidden puzzle in Sector 4

By far the easiest hidden entrance to spot. Starting from the Sector 4, solve the first puzzle presented (the one with the ball rolling down a slope and changing into the 3 primary colors) and soon you'll go up some stairs and find the second puzzle of the sector, pictured below.
 
Solve that by retracting both blue squares and getting to the top. Turn around and, with a not-so-tricky jump, get into that opening and touch the fake wall.
 
Once inside, wait for the steps in the center of the room to raise completely, then step on top of the button, which reveals the puzzle. The objective here is to shift the retractable rectangles using rotational wall buttons to align the resulting shapes with the tinted window near you. It's much simpler than it seems.

There are 3 parts to the chamber behind the window, with each of those being spinnable. Always go back and stand on top of the button in the center of the room to check the alignment between the tinted window and the shapes behind it.
 
Enough tips, here's how to solve it, starting from the part in the far back.
→ Background (#1 in the picture below): Rotate it twice and expand: top-left x2, top-right x3, bottom-left x3, bottom-right x2
→ Middle part (#2): Rotate it once counter-clockwise (to the right button) and expand: top-left x3, top-right x2, bottom-left x2, bottom-right x3
→ Foremost (#3): Rotate it twice and expand: right x2 and left x3.

Below you can see each part with only one possible move remaining to solve it (after properly rotating them).

If done correctly, as soon as you expand the last piece you'll hear a completion sound and the achievement pops up. Hurray!

Complete the hidden beta puzzle (Sector 5)

This puzzle and its achievement were also in the original version of the game. I recommend the guide below for some help on getting this GAWDDAMN ACHIEVEMENT.
http://sp.zhabite.com/sharedfiles/filedetails/?id=132973251

That hidden path also includes two other achievements: Behind the Scenes and Push Me To Completion.

Complete the hidden puzzle in Sector 6

Starting from the Sector 6, go and walk and sprint and run and listen to the plot for quite a bit of time until you get to the elevator, then here's some more walking because why not, and you'll find yourself some puzzles to solve. Three "loose wires" puzzles later, you'll arrive in a huge room, and what's unique about it is that it contains some blue balls of steel that you can touch but not really (it repells itself from you if you try to reach them).
 
In this room, what you need is right above the entrance. Go upstairs and soon you'll find a path of blocks on the wall that can be easily walked on and the fake wall right at the end. Go through it and you'll find the last hidden puzzle.
 
 
The objective here is to get the green balls into the green fields, which can be easily done by following the steps below.
 
1. Tilt the chamber with (A) (see picture above) to make the green ball that's nearest to the exit roll over onto the retracted red rectangle (z1) on the floor;
2. Expand said rectangle (z1) thrice so that the ball gets to the top part of the chamber;
3. Expand the red rectangle next to it (z2) so that you can lock up the ball in place;
4. Now tilt back with (B) twice;
5. Onto the next ball, after the second tilt you made before, it'll end up on the top of the red rectangle (y1). Expand it until you can't anymore, making it roll over next to (y2);
6. Back to the first ball, tilt the chamber to make it roll to its green field with the (C) button;
7. With one ball done, tilt the chamber back to flat with (D) and, finally...
8. Quickly expand the (Y2) rectangle, making the second ball roll over its designated field.
 
If you somehow mess up, there's always the restart button in front of you.

Congrats, you've got some new achievements. Everything achievement related here is the same from the original version of the game (as far as I can tell), except for the Tough Going parts 1&2 achievements that were made into only one in the Director's Cut.

For everything else, check out the original version's guide section.

There we go. If this guide has helped you out, don't forget to rate it. And if there is any way that I can make it clearer, do comment.