I'll bite. The best way to secure anything to brick is with a dedicated fastener, you'll need a 2 piece setup. Usually you drill a larger hole that accepts a plastic piece, and then the actual metal hook will go into that expanding the plastic piece to make it very snug.

They make special masonry bits you'll need to use to drill into the mortar if you don't have a hammer drill. Hammer drills vibrate when you drill, and are almost mandatory to drill into brick.

You can safely drill into the mortar with a regular drill and mortar bit (looks like has a spade on the end almost, just meant to take a beating). link to image of masonry bit:

http://www.lawson-his.co.uk/images/Heller/240956.jpg

example of fasteners:

http://fasteners.hardwarestore.com/learning/working-with-masonry-anchors.aspx

I wouldn't worry about sealing it, it should be like a 1/4" hole for the plastic, and should be very snug, water won't make much impact. But if you ever remove them, you should fill with something to keep water out.

Sorry, just reread your original message. You can use regular eye hooks, but probably longer threaded ones, and look for something a bit more durable than the in home versions, if you can find like galvanized or more "outdoor" capable.

The 2 piece anchor bit, just get the plastic piece from whatever "kit" home depot or lowes sells, and match your eye hooks to that, usually the masonry anrchor and screw come together (sometimes with a masonry bit as well so you know which sized bit to get). So just find a similar eye to replace the screws that came with the kit...

Really any kind of plastic anchor like the ones below will do. If you're not hanging a lot of weight, you can use smaller eye bolts.

http://www.naturalhandyman.com/iip/inffastener/i/plastic1.jpg

Here is another option

http://forums.gardenweb.com/forums/load/clematis/msg0311323612193.html

Thanks so much for your suggestions! I really appreciate the knowledge I've gained. I think that I'll try the trumax system first, because it is less invasive (no drilling). But if that isn't satisfactory, I'll go the anchor and eye screw route.