How to make "Update Check" to use a Web proxy?

Add a virtual server with a spare IP address and wildcard port. Add host entries on the F5 with the update server's names pointing to the spare address, this will override DNS and send the traffic to your virtual server. Set your pool members on the new virtual server as the proxy addresses.

this is mine.

HTTP Proxy Encapsulator v10/v11

 configuration

root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list sys global-settings remote-host
sys global-settings {
    remote-host {
        callhome.f5.com {
            addr 172.28.24.10
            hostname callhome.f5.com
        }
    }
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar
ltm virtual bar {
    destination 172.28.24.10:443
    ip-protocol tcp
    mask 255.255.255.255
    pool foo
    profiles {
        tcp { }
    }
    rules {
        qux
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 13
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm pool foo
ltm pool foo {
    members {
        172.28.24.1:3128 {
            address 172.28.24.1
        }
    }
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm rule qux
ltm rule qux {
    when CLIENT_ACCEPTED {
  set bypass 0
  set bufferdata ""
  TCP::collect 0 0
}
when CLIENT_DATA {
  if { $bypass eq 1 } {
    TCP::payload replace 0 [string length $bufferdata] ""
    TCP::release
    return
  }
  set bufferdata [TCP::payload]
  TCP::collect
}

when SERVER_CONNECTED {
  serverside {TCP::respond "CONNECT callhome.f5.com:443 HTTP/1.0\r\n\r\n"}
  TCP::collect
}

when SERVER_DATA {
  if { $bypass eq 1 } {
    TCP::release
    return
  }
  if { [TCP::payload] starts_with "HTTP/1.0 200 Connection established\r\n\r\n" } {
    TCP::payload replace 0 39 ""
    TCP::respond $bufferdata
    TCP::release
    set bypass 1
  } else {
    TCP::close
  }
}
}

 test

[root@ve11a:Active:In Sync] config  /usr/bin/updatecheck
[root@ve11a:Active:In Sync] config 

 trace

[root@ve11a:Active:In Sync] config  ssldump -Aed -nni 0.0 port 3128
New TCP connection 1: 172.28.24.14(51753) <-> 172.28.24.1(3128)
1419519138.7087 (0.0040)  C>S
---------------------------------------------------------------
CONNECT callhome.f5.com:443 HTTP/1.0

---------------------------------------------------------------

1419519139.1157 (0.4069)  S>C
---------------------------------------------------------------
HTTP/1.0 200 Connection established

---------------------------------------------------------------

1 1  1419519139.1157 (0.0000)  C>SV3.1(512)  Handshake
1 2  1419519139.3267 (0.2110)  S>CV3.3(81)  Handshake
1 3  1419519139.3292 (0.0024)  S>CV3.3(3657)  Handshake
1 4  1419519139.3292 (0.0000)  S>CV3.3(4)  Handshake
1 5  1419519139.3346 (0.0054)  C>SV3.3(262)  Handshake
1 6  1419519139.3346 (0.0000)  C>SV3.3(1)  ChangeCipherSpec
1 7  1419519139.3346 (0.0000)  C>SV3.3(80)  Handshake
1 8  1419519139.5468 (0.2121)  S>CV3.3(1)  ChangeCipherSpec
1 9  1419519139.5468 (0.0000)  S>CV3.3(80)  Handshake
1 10 1419519139.5754 (0.0286)  C>SV3.3(2144)  application_data
1 11 1419519139.8617 (0.2862)  S>CV3.3(1536)  application_data
1    1419519139.8690 (0.0073)  C>S  TCP FIN
1    1419519139.8697 (0.0007)  S>C  TCP FIN