The compilation error is strange. Either you already compiled all the objects (*.o files) or it did not even try to build them. It looks like $(OBJS) was not defined. Could you try to edit the Makefile and make put all the objects in one line, without those "\" at the end of line? Something like this (everything in one line):
OBJS = src/api/api_lib.o src/api/api_msg.o src/api/err.o src/api/netbuf.o src/api/netdb.o src/api/netifapi.o src/api/sockets.o src/api/tcpip.o src/core/ipv4/autoip.o src/core/ipv4/icmp.o src/core/ipv4/igmp.o src/core/ipv4/inet.o src/core/ipv4/inet_chksum.o src/core/ipv4/ip.o src/core/ipv4/ip_addr.o src/core/ipv4/ip_frag.o src/core/dhcp.o src/core/dns.o src/core/def.o src/core/init.o src/core/netif.o src/core/mem.o src/core/memp.o src/core/pbuf.o src/core/raw.o src/core/stats.o src/core/sys.o src/core/tcp.o src/core/tcp_in.o src/core/tcp_out.o src/core/timers.o src/core/udp.o src/netif/etharp.o espressif/espconn.o espressif/espconn_tcp.o espressif/espconn_udp.o espressif/sys_arch.o espressif/netio.o espressif/dhcpserver.o espressif/ping.o our/eagle_lwip_if.o
?