Slight follow-up. If you search for

DSolve ifun

in the Documentation Center and look at the various Tutorial links that come up and then, in them search for the string ifun to find the warning messages in some of the examples, you will see some cases where such messages arise and some discussion of the circumstances that they are responding to. Here's an example

http://reference.wolfram.com/language/tutorial/DSolveClassificationOfDifferentialEquations.html

I suspect that the message is being generated from the internal use of Solve when dealing with the presence of the square roots and alerting you to the fact that there may be an assumption in the answer as to the sign of the function.

If, for Real values, you allow y[x] to be assumed positive, then you can get a sense of this by transforming the equation in a way that removes a square root by y[x]==z[x]^2 and using the following equation:

D[z[x]^2, x] == ((2 Sqrt[y[x]] - 2 y[x])/x) /. y[x] -> z[x]^2

then

DSolve[2 z[x] Derivative[1][z][x] == (-2 z[x]^2 + 2 Sqrt[z[x]^2])/x, z[x], x]

yields (without an error message:

{{z[x] -> (E^C[1] + x)/x}}

and hence y[x] is

 (E^C[1] + x)/x}2