Best software for image segmentation for time-series images?

I don't know that there are any magic tricks for identifying regions of interest (ROI) that aren't available in your HCImage software. Although I'm not familiar with that particular software, it claims to allow for "object identification," which is what you're asking for.

The problem typically is less in the software itself than in the criteria you provide to the software for object identification. You need to provide an algorithm that makes a yes/no decision about whether a particular pixel belongs to any object and, if so, to which object it belongs. Then you use the outline of the object to specify the corresponding ROI.

That's very situation-specific. You typically have to specify criteria like size ranges, intensity and/or contrast ranges to determine boundaries, shape ranges, and so forth. You need to decide whether or how much you want to fill in an object for which some interior region doesn't seem to meet the criteria, and how to set the boundary if 2 objects are touching each other.

I suspect that almost all image-analysis software lets you set those criteria and to apply them automatically once you've set them, but there will be a process of trial and error to get the settings correct and you will still need to check visually whether the ultimate choices make sense. With fluorescence images this works best with high contrast between signal and background and objects that are physically separated. For example, it can be pretty easy to specify criteria for fluorescent nuclear labels, as the criteria can be relatively easy to specify. With your images* there's a fair amount of signal outside the nuclei and not a very strong signal-to-background ratio.

Advantages of ImageJ are that it doesn't tie you to a particular manufacturer's equipment, it has been developed over decades, many people have written add-ons to extend its basic functionality, and you can write your own analysis functions if you wish. But you will still have to do the work of showing the software how to analyze the types of images that you have.

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*These are necessarily 8-bit images here. If you have more dynamic range like 12-bit images, be careful because the way that the higher bit depth is mapped down to 8 bits can be misleading visually. So maybe you have more signal/background than is seen here.