What willroute4food is saying is that if you were to have 300 routers all with the TACACS source of 192.168.1.1 then the return packet from the TACACS server would never (well 1/300) get back to the right router.
Are you sure they didn't give you the whole /23 or something to use for loopbacks and to only use a /32 on each one for the loopback?
Correct, if you put this:
router eigrp 100
network 188.8.131.52 0.0.0.0
on every single router then your dicked (with the exact same ip on all your loopbacks). Thats exactly what Im saying. Now, if they gave you a /23 and said use a 32 bit mask for every address, then your ok. That snippet above, along with a /32 bit mask on the interface will only advertise a /32 (auto-summary be damned) to the network...and that is doable. But if what your saying is that you only have 1 ip address for 300 routers then your screwed from the git-go.