FOLLOWING ISSUE HAS ALREADY BEEN SOLVED
It seems EaglePlayer does NOT play the songs correctly in AmiKit. I found the reason of that and the fix is easy, fortunatelly. It's enough to decrease the mixing frequency from 44100Hz to 27500Hz.
Start EaglePlayer and select AmplifierManager from right-click menu. Then select 8bit Amplifier and popup its GUI by pressing Show button. Enable the Global option from right-click menu of this amplifier and then decrease the sample mixing frequency from 44100 to about 27500 Hz. (If the global option is enabled, the frequency is changed for all the channels). Do the same for 14bit Amplifier. Do not forget to save the settings of both amplifiers.
These amplifiers are used to mix the sound across the channels in order to get some sourround. However I've just found out that you can get a better sound quality (IMHO) by using classic 4ch output in EaglePlayer (without any mixing) while the crosschannel mixing could be done by WinUAE itself (by Sound -> Stereo Separation option). To test it, move the "4ch FastRAM Amplifier" to the top of the amplifier list in AmplifierManager, so every song is played by this amplifier then. Save the settings of EaglePlayer.
With regards to HippoPlayer (or any other Paula replayers), decrease every 44100 value found in its config to 27500 (except AHI ofcourse).