Squash blocks are for compressive loads on the joist, which your ceiling will not create. However, you can definitely screw to the flanges of your joists to fasten the framing to them. Just don't get carried away with oversized screws. Most manufacturers limit framing nails to 10d's, so that should give you an idea of what diameter screws are ok (I'm thinking #8'ish). I'd make every effort to avoid running screws into the connection between the flanges and the web...Put your screws into the meat of the flanges themselves. It is of course ok to screw into the web if you prefer to install blocking similar to squash blocks to facilitate attachment of your framing members.

 

Drop basement ceiling ubder TJI joists

Thanks, THEKCTERMITE - helps a bunch to talk with knowledgeable folks!!:thumbsup:

 

If the I-joist is a TJI, and only 1-1/2" thick with an engineered top and bottom chord, 8d nails are the maximum size to be used into those places. As Kc said, nailing into the web is recommended. If the chords are 1-3/4" or thicker, one (1), 10d nail is allowed into the top and bottom chord with 6"on center so as not to split the laminated chords.
This from installing blocking panels, face-nailing the top or bottom chords:

"Notes:
1. Details shown are for minimum blocking attachment. Transfer of lateral loads
may require additional fasteners. In such cases, nail size, spacing and specific
design detailing shall be provided by the building designer.
2. Common nails of the same pennyweight may be substituted for the box nails
shown above.
3. Where blocking is required between adjacent joists, the blocking panels may be
staggered by approximately 6 inches for LVL flange and 3 inches for lumber
flange, and end-nailed as shown in A, B, or C. If the blocking panels must be
kept in alignment because they are being used to provide diaphragm blocking for
the wood structural panels above, toenails may be used to hold them in place
until the diaphragm nailing is completed."
http://www.apawood.org/pdfs/managed/TT-078 .pdf?CFID=9561340&CFTOKEN=39282992

Be safe, Gary