I'm not sure what to suggest for why the connection sometimes fails. It may be because when you reset the board, the connection is started from the same source port again. You could try editing Client::connect() to use a random _srcport.
I have watched the connection by tshark, it does connect from a different port at each reset. When using the local echo example the connection looks like this;
3 17.126960 192.168.0.2 -> 192.168.0.3 ICMP Echo (ping) request
4 17.127116 192.168.0.3 -> 192.168.0.2 ICMP Echo (ping) reply
11 34.911229 192.168.0.2 -> 192.168.0.3 TCP 36588 > telnet [SYN] Seq=0 Win=5840 Len=0 MSS=1460 TSV=230615 TSER=0 WS=6
12 34.911382 192.168.0.3 -> 192.168.0.2 TCP telnet > 36588 [RST] Seq=1 Win=0 Len=0
It does respond to PING requests, but it does not respond to any telnet requests.
When using the client example, the tshark output looks like this;
72 520.957368 de:ad:be:ef:fe:ed -> Broadcast ARP Who has 192.168.0.2? Tell 192.168.0.3
73 520.957410 SamsungE_b0:47:e1 -> de:ad:be:ef:fe:ed ARP 192.168.0.2 is at 00:13:77:b0:47:e1
74 520.957554 192.168.0.3 -> 192.168.0.2 TCP [TCP Port numbers reused] blackjack > http [SYN] Seq=0 Win=2048 Len=0 MSS=1460
75 520.957605 192.168.0.2 -> 192.168.0.3 TCP http > blackjack [SYN, ACK] Seq=0 Ack=1 Win=5840 Len=0 MSS=1460
76 520.957755 192.168.0.3 -> 192.168.0.2 TCP blackjack > http [ACK] Seq=1 Ack=1 Win=2048 Len=0
77 520.995661 192.168.0.3 -> 192.168.0.2 TCP blackjack > http [FIN, ACK] Seq=1 Ack=1 Win=2048 Len=0
78 520.995811 192.168.0.2 -> 192.168.0.3 TCP http > blackjack [FIN, ACK] Seq=1 Ack=2 Win=5840 Len=0
79 520.995959 192.168.0.3 -> 192.168.0.2 TCP blackjack > http [ACK] Seq=2 Ack=2 Win=2048 Len=0
But when I check the Apache web server logs, there is nothing in there.