The typical swordsage lacks medium armor proficiency, but the ability AC bonus should work while wearing a mithral breastplate
A swordsage doesn't gain as a class feature proficiency with medium armor. However, the description of the special material mithral says
Most mithral armors are one category lighter than normal for purposes of movement and other limitations (for example, whether a barbarian can use her fast movement ability while wearing the armor or not). Heavy armors are treated as medium, and medium armors are treated as light, but light armors are still treated as light. (Dungeon Master's Guide 284)
This DM would rule—and I think most others would also—that these limitations include the swordsage's extraordinary ability AC bonus that applies—somewhat controversially—only while the swordsage wears light armor, therefore allowing a swordsage to gain the benefit of the special ability AC bonus while wearing, for example, a mithral breastplate.
Just in case there's some question, proficiency with light armor also includes proficiency with medium armors that have somehow become light armors, and, likewise, by extension, proficiency with medium armor includes heavy armors that have somehow become medium. While this can be inferred from most descriptions of special materials that make such changes to armor, there's also this Dragon #335 Sage Advice column “Official Answers to Your Questions” exchange:
Is a character proficient with light armor, such as a rogue, considered to be proficient with mithral breastplate? What about a character proficient with medium armor, such as a barbarian—is he considered proficient with mithral full plate armor?
The description of mithral on page 284 of the Dungeon Master’s Guide is less precise than it could be in defining how it interacts with armor proficiency rules. The simplest answer—and the one that the Sage expects most players and DMs use—is that mithral armor is treated as one category lighter for all purposes, including proficiency. This isn’t exactly what the Dungeon Master’s Guide says, but it’s a reasonable interpretation of the intent of the rule (and it’s supported by a number of precedents, including the descriptions of various specific mithral armors described on page 220 of the Dungeon Master’s Guide and a variety of NPC stat blocks).
Thus, a ranger or rogue could wear a mithral breastplate without suffering a nonproficiency penalty (since it’s treated as light armor), and each could use any ability dependent on wearing light or no armor (such as evasion or the ranger’s combat style). A barbarian could wear mithral full plate armor without suffering a nonproficiency penalty (since it’s treated as medium armor), and he could use any ability dependent on wearing medium or lighter armor (such as fast movement).
The same would be true of any other special material that uses the same or similar language as mithral (such as darkleaf, on page 120 of the Eberron Campaign Setting). (80)
This exchange is repeated nearly verbatim in the D&D Frequently Asked Questions (53–4), with which some have issues.
Proficiency with some armors made from the special material mithral is also specifically addressed by the Magic Item Compendium on Table 1–4: Mithral Armor in the table's footnotes that say that the particular medium or heavy armor is "treated as light [or medium] armor [respectively] for the purpose of movement, proficiency, and other limitations" (6).