I've extracted DNA using a kit and final step is eluted with buffer. If I must dilute my DNA samples, do I use the same elution buffer or can I use milliq water?
I would generally recommend using the elution buffer (which is typically Tris-EDTA buffer) or TE-Buffer, as the pH and the conditions stabilize the DNA for a longer time, which is especially interesting when you want to store your DNA.
It also dependent on your downstream application - when they recommend diluting the sample in water (or another buffer) I would follow this advice. In my experience TE buffer is perfect for all molecularbiology applications (cloning, digestion, amplification, PCR, etc.). The EDTA will of course complex bivalent cations, but since the concentration in the buffer is only 1mM and the DNA solution is usually diluted further in PCR or digests, this plays no role in real life applications.
The Quiagen kits I'm familiar with explicitly state that you can use water instead of elution buffer for elution. The choice of continuing to use buffer or water is yours and depends on what you want to do with the DNA and whether the buffer interferes with later steps. Using the buffer is probably safer for long-term storage, but I don't know how much this really matters (I never had any problem storing DNA in water).
If you dilute with water you'll obviously end up with a weaker buffer that might not be able to keep the pH at the specified value. So if you need your DNA at the elution buffer pH, you shouldn't dilute it.
Just to add to the above responses, you are better off not use MQ because if it is contaminated with DNases, then it will digest your construct if it reaches room temperature or 37 oC as you are melting the frozen construct in your hand! TE-buffer is the standard buffer for DNA elusions but make sure your column is compatible with it. For transfection based experiments TE works fine for me!