Try the command set arj_sw=-c before running the installation. ARJ skips the stamp check with its -c switch, and the environment variable is a way to set default switches.

The ykhwong SVN build supports changing date and time stamps on files, which I've seen working on Windows, but I'm not sure about other platforms.

I had exactly the same error like Great Hierophant reports. In fact everything that uses the same SSI installer has the problem described.
(EOB1 , EOB2 from EOB trilogy are affected, Ravenloft is affected and so on)
Haven't tried ripsaw8080's solution though, I always thought it was an installer issue or something.
BUT
I also found another case that dates play an important role: Realms of Arkania Blade of Destiny patch.
In the readme it says to copy the main executable (bladem.exe) from the installation floppy to the HDD dir and run their patcher. This does not work because of the date/time thingie. Patching on a real DOS machine works fine (original game ofcourse)

The weird part about this issue is that when I opened the ARJ files in 7zip, the file dates/times matched the files that were contained within multiple ARJ archive files.

^ I believe the problem is when the files get written on the HDD , the files don't preserve their original values and get changeds to whatever are the current ones based on the host system's ones

Does this happen with other Dosbox versions?

There is a DOS function to change the date and time stamp on a file that is often used by archive decompressor programs to set the stamp of extracted files to the stamp stored in the archive. DOSBox does not implement the function, partly due to cross-platform issues, but also because it's useful to know when files have been modified by programs running in the emulation. For example, an executable infector virus can't disguise itself by preserving the original stamp on an executable file that's been infected.

The EOB trilogy installs correctly with the environment variable mentioned in my previous comment. Note that the issue is not in the cdrom version, only in the floppy version where files are split across disks. The -c switch solution will probably work in any game installation that calls DEARJ.EXE, and you may get an indication that it's working when DEARJ displays a message that it's using the environment variable, as long as the output isn't redirected to NUL or so.

If I understand correctly, for the file in question, the decompressing utility decompresses the first part, writes that part to disk, and uses the DOS function to set the date/time stamps on the file to what it originally was (as stored within the archive). Then it asks for the next disk but checks to see if the decompressed file on the hard drive has the original file/time stamp. If it does, it continues decompressing the file. If it does not, it skips the file and goes onto the next file.

Thanks for the workaround.