#Mathematica <s>136</s> 80 bytes

This is a straightforward approach, working outwards from `n`.

`n` is a 7-distinct-prime product iff the number of prime factors is 7 (`PrimeNu@#==7`) and none of these factors appears more than once (`SquareFreeQ@#&`).


    g@n_:=(k=1;While[!(PrimeNu@#==7&&SquareFreeQ@#&)[z=n-Ceiling[k/2](-1)^k],k++];z)


My earlier submission (136 bytes) found both the first 7-distinct-prime product above `n` and, if it exists, the first 7-distinct-prime product below `n`. It then simply determined which was closer to `n`.  If the products were equidistant, it returned both.

The current version checks n-1,n+1,n-2,n+2... until it reaches the first 7-distinct-prime product.  This more efficient version adopts the approach Dennis took.

The key advance was in using `Ceiling[k/2](-1)^k]` to produce the series, 1, -1, 2, -2...

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    g[5]
    g[860782]
    g[1425060]

510510 

870870

1438710