There really isn't a good answer to this question. The problem is that your question constructs a dicotomy--are Jews white, or a race of their own? The answer really is both.

Jews do have strong genetic connections to the Middle East. Evidence is more conflicted for particularly far-flung communities, and exactly how Middle Eastern is a matter of contention, but the basic premise holds true--Jews from diverse areas share Middle Eastern ancestry.

However, there's no set boundary for when a group of people is sufficiently distinct to be "a unique race". Jews also are genetically related to the people they live around, the genetic separation isn't complete.

"Race" in general is a somewhat frought phenomenon anyway. On US Census forms Arabs and other Middle Eastern people are theoretically supposed to check "white", yet in popular discourse they're usually not termed white people. On the other hand, there are groups, such as [Ashkenazi] Jews, who were once thought of as non-white but now are usually thought of as white. No genes changed, it was simply the social conceptions of what makes up a racial group.

So the question really isn't "are Jews a unique race or white". Britons are different genetically than Italians, yet both are white in modern American perception. Similarly, Jews are genetically related, but many are white, simply because white in the modern American imagination is defined that way.

As an aside, it's important to note that no matter the definition of white, not all Jews are white. Most Jews are, worldwide, but not all by a long shot (depending, again, on your definition of white). In America the overwhelming majority of Jews are Ashkenazi, from Central or Eastern Europe, and are thought of as white. However, worldwide many Jews are Middle Eastern (with widely varying skin color), and may or may not be thought of as white in the US. There are a significant number of Ethiopian Jews, who are black, and there is a population of Indian Jews. Not to mention, conversion and intermarriage in the US results in Jews with a huge variety of ethnic backgrounds, even if the overwhelming majority of Jews in the US are Ashkenazi, and therefore white.

tl;dr Jews are an ethnic group. Many Jews are white. Those two things are largely independent.