Although single-coloured ensigns started replacing the old Tudor stripped ensigns since the early 1620s, it wasn't before the First Dutch War that the practice of supplying all the ships of a squadron with ensigns of the squadronal colour (red, white and blue) was ultimately established.
As for the White Ensign, it remained the same until 1702, when it was shortly modified to add a large Saint George's Cross centered in the field (see image at left). This and other versions were replaced in 1707 by the new United Kingdom white ensign, this including the Union Jack at the corner.
Sources: Anonymous manuscript from 1670, reprinted in Amsterdam 1966 as «Flags of the World 1669-1670» | W. G. Perrin, «British flags», Cambridge 1922
Sizes: ensign W12 x L18 mm, mast flags W8 x L12 mm, jack W6 x L9 mm, pennant W4 x L48 mm.