  We present multi-epoch mid-infrared (IR) photometry and the optical discovery
observations of the "impostor" supernova (SN) 2010da in NGC 300 using new and
archival Spitzer Space Telescope images and ground-based observatories. The
mid-IR counterpart of SN 2010da was detected as SPIRITS 14bme in the SPitzer
InfraRed Intensive Transient Survey (SPIRITS), an ongoing systematic search for
IR transients. A sharp increase in the 3.6 $\mu$m flux followed by a rapid
decrease measured ~150 d before and ~80 d after the initial outburst,
respectively, reveal a mid-IR counterpart to the coincident optical and high
luminosity X-ray outbursts. At late times after the outburst (~2000 d), the 3.6
and 4.5 $\mu$m emission increased to over a factor of 2 times the progenitor
flux. We attribute the re-brightening mid-IR emission to continued dust
production and increasing luminosity of the surviving system associated with SN
2010da. We analyze the evolution of the dust temperature, mass, luminosity, and
equilibrium temperature radius in order to resolve the nature of SN 2010da. We
address the leading interpretation of SN 2010da as an eruption from a luminous
blue variable (LBV) high-mass X-ray binary (HMXB) system. We propose that SN
2010da is instead a supergiant (sg)B[e]-HMXB based on similar luminosities and
dust masses exhibited by two other known sgB[e]-HMXB systems. Additionally, the
SN 2010da progenitor occupies a similar region on a mid-IR color-magnitude
diagram (CMD) with known sgB[e] stars in the Large Magellanic Cloud. The lower
limit estimated for the orbital eccentricity of the sgB[e]-HMXB (e>0.82) from
X-ray luminosity measurements is high compared to known sgHMXBs and supports
the claim that SN 2010da may be associated with a newly formed HMXB system.
