here's my take on dealing with mdf/partical board. first off what size did you buy? you said they were out of 5/8". did you get 1/2" or 3/4". if you got 1/2" options 1 and 3 will be out.
option 1:
if you got 3/4, screwing through the edges is possible without splitting the wood, you just need to be very careful and hit the center. i built a simple little jig to help with the centering of the holes when i do this. drill a pilot hole first then put in the screw. you need to realize what splits the wood isn't the screw, it's the wood your displacing with it. plan on screwing the pieces together once, eveytime you take out the screw the joint will weaken. also spread wood glue across the edges between the joints and in the pilot holes to add strength.
easiest and weakest way to go.
option 2:
use a corner brace. this is similar to using L barckets, but stronger. get some 1x1 stock to run along the inside corners and screw into this instead of the mdf. attatch it completely to one side first, then align the second side up, clamp it together, then screw. pilot holes will make things easier. again use wood glue on all wood to wood conections. this is a more time consuming method but will give you much better joints and the only extra tools needed is some clamps.
option 3:
use a router/biscuit jointer and biscuits. once you've notched the edges and ran a bead of glue, slide in the biscuits, and clamp the sides together. i would still use a few braces, but it's not as important for it to run the length of the joint. just put small ones at the beginning and ending of the joint, if it's a long one brace it in the middle as well. this will make a rock solid joint, but you will need the tools do it.
option 4:
build a frame for your cabinet first. you could use cheap 2x4's. then attatch the mdf. this is a solid way to go, but will require some framing skill.
personally i'm going to use a combination of 2 and 4. frame the top and bottom of the cabinet(and of course the monitor area) and use braces for the rest. hope this helps.