It depends on the type of analysis you are planning and how irregular your cells are. Anytime pixels are forced to become square, some amount of resampling and deformation is going to take place. You would need to provide more information about the satellite sensor your data originated from and what you're going to be doing with it in ArcGIS before I could tell you if it's appropriate for what you're doing.
But, ArcGIS will force raster pixels to be square upon import anyway. For MODIS swath data, this can significantly change the extent of the layer and make results of many analyses completely unreliable, even if not projecting the data. If you started out with square pixels, projecting may change the shape, and forcing them to be square upon reprojection will involve resampling or deformation. For a Lambert to UTM reprojection, any distortion or resampling shouldn't have a large effect on results.
What I normally do is keep everything in feature format. I.e. represent each cell by the centroid point, and use the Thiessen polygon tool to create polygons for each cell after projecting the points to the new coordinate system if I need them. This also makes it possible to use certain tools in ArcGIS that won't work with raster data, allows you to store more attributes for each point and do quick calculations using the reflectance values in each of the bands just through the attribute table, and can lead to speed improvements for certain analyses.