The Institute of Heraldry does not have any publicly available records regarding the history or symbolism of the 282nd Support Battalion Distinctive Unit Insignia. Likewise, the U.S. Army Center of Military History, online repository of lineages and honors for hundreds if not thousands of unit, has no record of the Battalion.
According to the U.S. Army Europe’s Community Command Plan for 1991, the 282nd Support Battalion—also known as the 282nd Base Support Battalion (BSB)—was originally designated as the 92nd Support Battalion. The 282nd (BSB) and was assigned to the 100th Area Support Group and headquartered at Hohenfels.
After the turn of the millennium, the U.S. Army began the process of reflagging the Area Support Groups, Base Support Battalions, and Area Support Teams to mirror the structure of Army Garrisons around the world, i.e., Standard Garrison Organization. As a consequence of this realignment, the 100th Area Support Group, its component Base Support Battalions (281st, 282nd, 283rd, and 401st), and the Battalion’s sub-units were inactivated and reflagged as U.S. Army Garrison Hohenfels in 2005.
Regarding the symbolism and imagery of this insignia: the checkerboard seen in the chief of the insignia is an image usually associated with Military Intelligence, but it is also sometimes used to symbolize teamwork and cooperation. The maroon chevron is a sign of support, and the motto “Support The Pillars” is obviously a reference to the four columns the lion passant—which appears to be taken from the Coat of Arms of England—is navigating.
Full guidance on wear of the DUI is found in DA Pamphlet 670-1,
Section 21-22, "Distinctive unit insignia" and 21–3(d) and (e),
"Beret" and
"Garrison Cap," respectively.
(H/T to the U.S. Army in Germany Web site and its "Military Communities" page)