Take a look a the "createmeta" for your project. You can retrieve it by making a GET request to <your_jira_server>/rest/api/2/issue/createmeta?expand=projects.issuetypes.fields&projectIds=<project_id>
That should give you more detailed information about the expected format for the data your field. With checkboxes, you'll generally find something like:
"customfield_10600": {
"required": false,
"schema": {
"type": "array",
"items": "option",
"custom": "com.atlassian.jira.plugin.system.customfieldtypes:multicheckboxes",
"customId": 10600
},
"name": "My Checkbox",
"key": "customfield_10600",
"hasDefaultValue": false,
"operations": [
"add",
"set",
"remove"
],
"allowedValues": [
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10400",
"value": "apples",
"id": "10400"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10401",
"value": "bananas",
"id": "10401"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10402",
"value": "grapes",
"id": "10402"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10403",
"value": "kiwi",
"id": "10403"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10404",
"value": "limes",
"id": "10404"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10405",
"value": "oranges",
"id": "10405"
},
{
"self": "<your_jira_server>/rest/api/2/customFieldOption/10406",
"value": "pears",
"id": "10406"
}
]
}
When you're sending it back to JIRA, it expects an array of option types. For options, you should be able to use either a {"name": value} or {"id": id} JSON object (I've only ever used the ID approach). You should use the value or id from the list of allowed values.
If you want to set more than one, you'll need to send an array of those option objects.