Yes you can use the 3560 to route the traffic. ( pls see info below )
create 2 VLANs for each subnet & then create 2 L3 interfaces with IPs 192.168.2.1 & 192.168.11.1 on the switch.
***
You are not able to browse the internet from the new subnet because The new subnet may not have been configured for NAT on the netscreen box.
***
Cisco Catalyst 3560 Series switches can be purchased with the IP Base (formerly called SMI) the field to Advanced IP Services licenses.
The IP Base license offers advanced QoS, ratelimiting, ACLs, IGMP Snooping, and routing capabilities such as RIP, static routing, and EIGRP Stub.
HTH - Pls rate all useful posts