i've never made one like this but I totally know what I'm talking about (for real, so trust me)
A pneumatic screen door closer PULLS the door closed with an internal spring. An advantage is a little screw valve on the end of the tube which regulates how fast escaping air moves the piston, & thus how fast the door closes. Note: I have a Stanley pneumatic closer on my gate which is affected by temp & humidity. When it's cold & damp it won't close all the way, so I have to open the air screw, but when the weather is dry, that setting slams the gate. Indoor might be different. The clock gizmo (below, sans shock absorber & 2x2) can be mounted someplace convenient w/ a cord for door restraint. Make some kind of latch to coon-proof the door when closed, and a way to keep straw, poop, & feathers from gumming up the door's slide channel. It'll be a somewhat different configuration from the Gearloose Model A, but some parts will transfer. The width of the door is limited by the length of the piston unless you use a lever of some sort (like a 2x2, see Gearloose Model A).
This one below works like a sliding camera shutter. The door frame is open at the bottom so straw & stuff won't prevent the door from closing (ok anything is possible, but it's like some percent less likely if there's no place for junk to accumulate).
The top piece of the grey alarm shut-off lever has a enough leverage to crush the clock but only requires a light tap, so it could probably be made of a piece of rubber hose for safety sake.
The little red dot marked "X" is a headless nail, for marking location of the right pressure of the blue "pelican hook" (secondary trigger) for placement under the primary trigger (lavender). in the other drawings (which I'm not going go back & change) it's on the right side of the pelican hook (or below it in the gearloose drawings), but it works better on the left (above) because when the secondary trigger is set & under pressure, it can't slip up the angle of the primary towards the clock, so the primary trigger can be straight (instead of previously bent w/ a bracket to keep it from tipping (because It's bent)) which requires less tinkering.
The small psychedelically colored rectangles above represent spacer blocks to set the whole gizmo out from the wall because the shutter type door (as drawn here) won't fit between studs on 24 inch centers, so you'll probably have to frame the chicken door w/ a sill & all unless you fore go standard spacing, but a Sawz-all is great for some kinds of remodeling.
Here's the same one with Parallel Levers to tip the door so it fits between studs on 24 inch centers. I kinda like this one but it's too complicated (fun to draw though).
The Gearloose model "A" with shock-absorber/guillotine-door is a lot simpler.