I'm working to find a death date for Samuel. According to the Census records, it appears he died sometime between 1861 and 1871.
The 1861 Census lists the family but, for some reason, Samuel is listed as "James". The surname is Heyes and his occupation is identified as "Pensioner" so I presume that it's still Samuel despite the error.
The 1871 Census appears to show that Eliza Jane has remarried to a Thomas Bate, a Railway Official who was born at Northwich, Cheshire. John Hayes, Samuel's and Eliza's youngest son, is living with his mother and step-father. He's the only other person in the household.
GRO lists the death of a Samuel Hayes in Warrington District in 1864 which is probably him:
HAYES, SAMUEL 71
GRO Reference: 1864 J Quarter in WARRINGTON Volume 08C Page 95
This date jives with the record of Eliza Jane remarrying:
Marriage: 30 Sep 1865 St Paul, Warrington, Lancashire, England
Thomas Bate - 62, Railway Policeman, Widower, Heath Street
Eliza Jane Hayes - 51, Widow, Silver Street
Groom's Father: Richard Bate, Labourer
Bride's Father: Henry Searle, Labourer
Witness: John Massey, (X); Jane Bruce, (X)
Married by Banns by: John D. Massingham
Register: Marriages 1865 - 1869, Page 18, Entry 35
Source: Original Parish Register