LipidTox stains live cells nicely, and the lipid droplets are stable for a few days as long as the cells are healthy. There will be basal uptake of the stain into non-lipid droplet containing cells that will stain some other compartments to a lower level (cytosol a bit as well), but overall you should be able to distinctly tell the difference between lipid droplet positive and negative cells. The difference is highly apparent by widefield fluorescence microscopy, so it should also be clear by flow cytometry.
Staining the cells with 1:1000 dilution of LipidTox Red or Green for 30 minutes at 37 °C is sufficient.