Arthur Collins' "The baronettage of England; ..." Vol 1 lists the children of Sir Rowland as 3 daughters (Sibil, Elizabeth, Mary) and 6 sons (Thomas, John, Arthur, Philip, Rowland and Charles) but no mention of dates/places of birth.
Casting around on line there's an entry on Freereg (transcription) that records the baptism of a Rond Egerton, 14 Dec 1619, Audley, Staffs. FindMyPast has the image (transcription = Rondus, father's given name Radi, no surname, date 14 dec 1619). As luck would have it the image is not as clear as one would wish.