We present a search for [C II] line and dust continuum emission from optical dropout galaxies at z > 6 using ASPECS, our Atacama Large Millimeter submillimeter Array Spectroscopic Survey in the Hubble Ultra-deep Field (UDF). Our observations, which cover the frequency range of 212-272 GHz, encompass approximately the range of 6 < z < 8 for [C II] line emission and reach a limiting luminosity of L-[C II] similar to (1.6-2.5) x 10(8) L-circle dot.We identify 14 [C II] line emitting candidates in this redshift range with significances > 4.5 sigma, two of which correspond to blind detections with no optical counterparts. At this significance level, our statistical analysis shows that about 60% of our candidates are expected to be spurious. For one of our blindly selected [C II] line candidates, we tentatively detect the CO(6-5) line in our parallel 3 mm line scan. None of the line candidates are individually detected in the 1.2 mm continuum. A stack of all [C II] candidates results in a tentative detection with S-1.2 (mm) = 14 +/- 5 mu Jy. This implies a dust-obscured star-formation rate (SFR) of (3 +/- 1) M-circle dot yr(-1). We find that the two highest-SFR objects have candidate [C II] lines with luminosities that are consistent with the low-redshift L-[C II] versus SFR relation. The other candidates have significantly higher [C II] luminosities than expected from their UV-based SFR. At the current sensitivity, it is unclear whether the majority of these sources are intrinsically bright [C II] emitters, or spurious sources. If only one of our line candidates was real (a scenario greatly favored by our statistical analysis), we find a source density for [C II] emitters at 6 < z < 8 that is significantly higher than predicted by current models and some extrapolations from galaxies in the local universe.