Military satellite reception is very elusive and over 99% is encrypted, except for occasional Brazilian pirates using US satellites. Otherwise its probably the easiest to pick up using a dongle receiver and an easy to make antenna.
The basic frequency range is about 245 to 270MHz, well within an RTL dongle's capability. The Brazil pirates seem to use the same frequency of 255.55MHz so they are easy to find. The satellites use circular polarized antennas but its fairly easy to create circular polarization. I think a pair of cross dipoles fed 90deg out of phase and about .25 wavelength over a ground plane would be a good basic directional antenna with some gain and circular pol.
You could start with a hardware cloth ground plane about 24" square, then put a galvanized steel 1/2" pipe flange in the middle of it. Then get a PVC pipe threaded adapter for 1/2" PVC pipe and glue a length of 1/2" PVC pipe in so the whole thing is about 10" long when threaded into the pipe flange and sticking out from the hardware cloth.
Then you need a pair of dipoles which you can make out of #12 house wire or 1/8" welding rod, etc. Each element will be about 10 1/2" long and you can drill small holes about 1/4 down from the end of the PVC pipe to hold the wire tightly. Looking down at the end of the PVC pipe the elements should look like + sticking out of the sides of the pipe.
Then you need some 50 ohm coaxial cable as a feedline connected to one pair of dipoles and a short run of RG-58 should be fine. Then you would need about 8.5" of 75 ohm coax as a delay line feeding the other dipole to create circular polarization and to feed the other dipole, RG-59 or RG-6 would work. The 75 ohm cable would connect to the junction of the 50 ohm coax on one pair of dipoles then to the other dipole.
The hard part is you would need to first tune each dipole to resonate around 260MHz with a 50 ohm cable attached, then trim the 75 ohm cable to 260MHz using an antenna analyzer or similar. The lengths can be calculated somewhat but bare ends of coax flared out to reach the dipole elements will skew the calculated length some at these frequencies.
Another hurdle is configuring the polarity of the coax that receives only the 75 ohm coax so you end up with Right Hand Circular Polarity. Get it right and the antenna will receive great. Get it backwards and it probably won't pick up anything. The only way to get it right is to monitor some rare satellite signals and swap the coax polarity and go with what works best. Or have someone make the antenna I described, find the proper way to connect the coax and post the results.
I might be able to do this but no guarantee on when. If you do end up perusing something like this you will be demonstrating an antenna that has directional properties, gain and circular polarity. Plus picking up signals from around 22, 300mi in space. BTW the Brazilian pirates are using a satellite at either 100deg W or 105deg W and from your approximate location (Baltimore?) the look angle would be around 38deg elevation and 215deg azimuth for the 100deg slot or 36deg elevation and 220deg for the 105 slot. With the antenna described you could easily point at one orbital slot and pick up the other since the antenna has a fairly wide beam.
prcguy